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Theorem fmptapd 7175
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)
Hypotheses
Ref Expression
fmptapd.a (𝜑𝐴𝑉)
fmptapd.b (𝜑𝐵𝑊)
fmptapd.s (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.c ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
Assertion
Ref Expression
fmptapd (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptapd
StepHypRef Expression
1 fmptapd.c . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
2 fmptapd.a . . . 4 (𝜑𝐴𝑉)
3 fmptapd.b . . . 4 (𝜑𝐵𝑊)
41, 2, 3fmptsnd 7173 . . 3 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
54uneq2d 4122 . 2 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6 mptun 6685 . . 3 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
76a1i 11 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8 fmptapd.s . . 3 (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
98mpteq1d 5203 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
105, 7, 93eqtr2d 2806 1 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401   = wceq 1570  wcel 2146  cun 3904  {csn 4591  cop 4597  cmpt 5194
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-mpt 5195
This theorem is used by:  fmptpr  7176  poimirlem3  38333  poimirlem4  38334  poimirlem16  38346  poimirlem17  38347  poimirlem19  38349  poimirlem20  38350
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